The use of mathematics to read the book of nature: About Kepler and snowflakes

Abstract

“Philosophy is written in that great book which ever lies before our eyes – I mean the universe – but we cannot understand it if we do not first learn the language and grasp the symbols, in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles and other geometric figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth,” wrote Galileo (Il Saggiatore, chapter 6, p. 4). In 1611, the mathematician Johannes Kepler,a contemporary of Galileo and voracious reader of the book of the world, wrote his shortest and most surprising book, The Six-Cornered Snowflake: A New Year’s Gift. “Even as I write these things, it has begun to snow again, and more thickly than before. I have been attentively observing the tiny particles of snow, and although they were all falling with pointed radii, they were of two kinds. Some were exceedingly small, with varying numbers of radii that spread in every direction and were plain, without tufts or striations. These were most delicate, but at the same time joined together at the center in a somewhat larger droplet; and they were the majority. Sprinkled among them were the rarer, six-cornered snowflakes” (Kepler, 1611). This text by Kepler, little known outside the physics and mathematics community, marked a milestone in the use of mathematics to understand a part of the physical world that surrounds us. With this text as a map, this article covers part of the terrain explored by geometry, from the 3rd century AD until today.